It is often useful to describe spatial geometry not by the spatial metric but by a triad which defines three vector fields which are orthogonal to each other and normalized in each point. This yields all information about spatial geometry, and indeed the inverse metric is obtained from the triad by where we sum over the index counting the triad vector fields. There are differences, however, between metric and triad formulations. First, the set of triad vectors can be rotated without changing the metric, which implies an additional gauge freedom with group SO(3) acting on the index . Invariance of the theory under those rotations is then guaranteed by a Gauss constraint in addition to the diffeomorphism and Hamiltonian constraints.
The second difference will turn out to be more important later on: We can not only rotate the triad vectors but also reflect them, i.e., change the orientation of the triad given by . This does not change the metric either, and so could be included in the gauge group as O(3). However, reflections are not connected to the unit element of O(3) and thus are not generated by a constraint. It then has to be seen whether or not the theory allows to impose invariance under reflections, i.e., if its solutions are reflection symmetric. This is not usually an issue in the classical theory since positive and negative orientations on the space of triads are separated by degenerate configurations where the determinant of the metric vanishes. Points on the boundary are usually singularities where the classical evolution breaks down such that we will never connect between both sides. However, since there are expectations that quantum gravity may resolve classical singularities, which indeed are confirmed in loop quantum cosmology, we will have to keep this issue in mind and not restrict to only one orientation from the outset.
© Max Planck Society and the author(s)